Exotic operators on the Hardy spaces of the infinite-dimensional torus

Abstract

We quantify the failure of interpolation between Hardy spaces $H^p({\mathbb{T}}^{\infty })$ on the infinite-dimensional torus. For any set $A\subseteq [1,\infty ]$ such that $A\setminus \{\infty \}$ is closed in $[1,\infty )$, we show that there exists an operator that is densely defined on $H^p({\mathbb{T}}^{\infty })$ for all $1\le p<\infty$ and weak-$\ast$ densely defined on $H^{\infty }({\mathbb{T}}^{\infty })$ that extends to a bounded linear operator on $H^p({\mathbb{T}}^{\infty })$ if and only if $p\in A$.

11 pages